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A probability of encounter with interstellar comets and the likelihood of their existence
ICARUS 27, 123--133 (1976)
A Probability of Encounter w i t h Interstellar C o m e t s and the Likelihood of their Existence ZDENEK SEKANINA Center for Astrophysics, Harvard College Observatory and Smithsonian Astrophysical Observatory, Cambridge, Massachusetts 02138 l~eceived May l, 1975; r e v i s e d J u n e 26, 1975 A t h e o r y o f t h e p r o b a b i l i t y o f e n c o u n t e r o f t h e Sun w i t h an i n t e r s t e l l a r c o m e t a t a d i s t a n c e c o m p a r a b l e to t h e E a r t h - S t m d i s t a n c e is f o r m u l a t e d , a n d a general e x p r e s s i o n is d e r i v e d e s t a b l i s h i n g t h e r e l a t i o n s h i p a m o n g t h e influx r a t e o f interstellar c o m e t s , t h e perihelion d i s t a n c e , t h e space d e n s i t y o f t h e c o m e t s , t h e M a x w e l l i a n d i s t r i b u t i o n o f c o m e t velocities in t h e i n t e r s t e l l a r cloud, a n d t h e c l o u d ' s s y s t e m a t i c v e l o c i t y relative t o t h e Sun. The f a c t t h a t no c o m e t w i t h a s t r o n g l y h y p e r b o l i c o r b i t h a s so far b e e n o b s e r v e d is u s e d t o d e t e r m i n e a n u p p e r limit o f 6 × l 0 -4 solar m a s s e s p e r cubic parsec (4 × 1 0 - 2 6 g c m -3) for t h e space density of interstellar comets. The theoretical distribution of semimajor axes of i n t e r s t e l l a r c o m e t s is d e r i v e d t o s h o w t h a t a s t r o n g h y p e r b o l i c excess m u s t be p r e s e n t in t h e o r b i t s o f a m a j o r i t y of i n t e r s t e l l a r c o m e t s regardless of t h e d y n a m i c a l c h a r a c t e r i s t i c s o f t h e c o m e t cloud, e x c e p t w h e n t h e cloud is m o v i n g along w i t h t h e Sun a n d t h e d i s t r i b u t i o n o f i n d i v i d u a l velocities has a v e r y low dispersion. This case, h o w e v e r , implies a p o s s i b i l i t y o f c a p t u r e b y t h e Sun a n d t h u s b e c o m e s a p r o b l e m o f a n O o r t - t y p e cloud. I . INTRODUCTION
The strong dynamical evidence corroborating Oort's (1950) hypothesis of a cometary cloud permanently associated with the solar system has recently been further reinforced by Marsden et al. (1973), who show t h a t the proper account of the nongravitational effects in the motions of nearly parabolic comets leads to smaller dimensions of the Oort cloud and therefore to its stronger gravitational bonds with the solar system, and by Marsden and Sekanina (1973), who find t h a t the aphelion distances of nearly parabolic comets least affected by the nongravitational forces pile around 50 000AU. On the other hand, some of the recent accomplishments in the physics of comets - - i n particular, the discovery in Comet Kohoutek (1973f) of methyl cyanide (Ulich and Conklin, 1974) and hydrogen cyanide (Huebner et al., 1974), two species previously detected in interstellar space-suggest t h a t more attention should be paid in the future to the relationship between Copyright © 1976by AcademicPress,Inc. All rights of reproduction in any form reserved. Printed in Great Britain
comets and interstellar matter. Although the discovery of interstellar molecules in comets does not offer any straightforward interpretation in terms of the interaction between comets and interstellar matter, it may, at least temporarily, place the theories of interstellar comet origin into a more favorable light, especially when assisted by some previously known circumstantial evidence [such as the anisotropic distribution of comet-aphelion directions; see, e.g., Tyror (1957) and Hurnik (1959)]. The aim of this paper is to investigate the possibility of the existence of a cloud of interstellar comets in the Sun's neighborhood. The study is based on a new approach to the relationship between the space density of interstellar comets and their distribution of velocities in the cloud, on the one hand, and their rate of influx to the observable region of the solar system and the distribution of their orbital elements on the other. Some work along these lines was done in the past [for a brief review of the research up to about a decade ago, see Richter (1963, pp. 147-152)], hut
123
124
ZDENEK S E K A N I N A
the results seem to d e p e n d r a t h e r strongly on the starting assumptions. The present a p p r o a c h is f o r m u l a t e d with a m i n i m u m of necessary restrictions; y e t it is mathematically t r a c t a b l e with relative ease. II. GENERAL EXPRESSION FOR THE INFLUX R A T E OF INTERSTELLAR COMETS
As in a n y other class of objects in the Galaxy, the members of a cloud of interstellar comets m u s t have a certain velocity distribution relative to their center of mass. I f the gravitational interactions a m o n g the individual objects of the class a n d the p e r t u r b a t i o n s from the Sun and o t h e r a m b i e n t objects are neglected, the stellar statistics show t h a t the normalized velocity distribution within a n y such class of objects can be a p p r o x i m a t e d b y the Gaussian distribution function ¢ ( v x, vy, v~) dv x dv r dv z = (27r) -3/2 (axCry(yz)-1
i_[fVx: r_x)2
× exp |
+
[
2ax2
+
v )2
2at2
( V z - V~)2]i
]l VxdV dV ,
where the directions of the three axes of the reference system of coordinates are det e r m i n e d b y the vectors x, y, z. The velocity distribution is defined b y six constants: Vx, Vr, V~ are the c o m p o n e n t s of the velocity o f the cloud's center o f mass relative to the Sun, and a~, %, a~ are the c o m p o n e n t s of the s t a n d a r d velocity dispersion in the cloud. Then, ¢(vx, vr, Vz) dvx dvy dv~ is numerically the relative fraction (percentage) of comets in the cloud, for which the velocity components relative to the Sun lie between v~ and vx ÷ dvx, vr and vr + dvy, and v~ and Vz + dvz. To reduce the n u m b e r of u n k n o w n s and to facilitate the integration substantially, we assume a spherical r a t h e r t h a n an ellipsoidal distribution of velocities; i.e., we define ax = % = a~ - ~.
(2)
This assumption also allows us to identify, w i t h o u t imposing a n y f u r t h e r restrictions, the direction of m o t i o n of the cloud's center
of mass with a n y one of the three coordinate axes. W e therefore define a new system of coordinates (x, y, z), whose center is located in the Sun and whose +z axis directs along the v e c t o r Y of the systematic velocity of the center of mass of the comet cloud relative to the Sun. I f U is the vector of an individual relative velocity referred to the Sun, and u, v, w are its c o m p o n e n t s in the three axes, distribution (1) becomes ~5(u, v, w) du dv dw = (2~r)-3/2 ~-3 × exp [ - ( U 2 + V 2 _ 2Vw)/2a 2] × du dv dw.
(a)
We recall t h a t ~5 is the distribution o f c o m e t a r y velocities before a n y appreciable gravitational pull of the Sun has commenced. N e x t we consider an e l e m e n t a r y surface area d S located at (xo, Yo, %) on a sphere o f radius r centered on the Sun. I f N o is the spatial concentration o f comets (per unit volume), assumed to be constant throughout the cloud, and is(U, v, w, Xo, Yo, %) is the flux rate of comets at d S (per unit area per unit time) at velocity U = (u, v, w), t h e n the n u m b e r of interstellar comets t h a t pass t h r o u g h d S per unit time with velocities between U and U + d U and with velocity c o m p o n e n t s between u and u + du, v and v + dv, and w and w + dw, respectively, is is(U, v, w, x o, Yo, Zo) cos ~bd S du dv dw = No r2 U ~(u, v, w) × cos~bsin¢d$dAdudvdw,
(4)
where ~b is the angle between the velocity vector U and the direction from d S t o w a r d the Sun, ¢ is given b y (3), and the angles and A are the polar coordinates of d S defined b y (Xo)[rsinqbc°sh\ Yo = | r s i n ~ b s i n A ] . zo \ r cos~b ]
(5)
N e x t we t r a n s f o r m the velocity components u, v, w to a n o t h e r system of coordinates, which has its center in the elem e n t a r y area dS, its + Z axis directed along the prolonged radius v e c t o r a w a y from the Sun, and its X and Y axes t a n g e n t to the
125
ENCOUNTERS WITH INTERSTELLAR COMETS
sphere of radius r. F u r t h e r , ~ is the angle from the X axis a r b i t r a r i l y oriented in the t a n g e n t plane, and 7 is the angle c o u n t e d from the + Z axis. The velocity c o m p o n e n t s u, v, w are now related to those of the new system as follows.
(:)
co i)(i 0
0/
=|--COS)t sin2 cos¢ sincb w 0 - s i n ¢ cos ~b/ [ U sin 7 e o s ~ x | V sin 7 s i n $ } , (6)
and cos 7 -->-1. T h e p r o d u c t r sin 7 converges to r sin ~ = lira q
+ z
+ z
r-~ oo
= qz - m
+z
,
(10)
so t h a t
-~z ~ U r ~°71 = l i m K - ~ 2 +
(~
\ Ucos~ / ×
and au Ov aw aU aU OU au av aw
Ou Ov 0~1 O7
-bz
q'-
U z sin 7.
=~--(]+qz) 2qz
(7)
3w 07
This t r a n s f o r m a t i o n allows us to i n t e g r a t e (4) over all possible angles $ and to write the influx o f the interstellar comets with velocities between U and U-4-dU and in directions between 7 and 7 + d~ (90 ° < 7 -<180 °) t h r o u g h dS per unit time:
+z
(11)
I f we integrate (8) over the whole sphere and allow r -~ ¢c, the influx rate (per unit time) o f interstellar comets with original perihelion distances between q and q + dq and original semimajor axes between z and z + dz becomes
i,(q, z) dq dz = ~2~)1/2 No K3 (Va) -~ × exp (-- V:
i~s(U, 7, Xo, Yo, %) cos ¢ dS d U d 7 = (27r)-l/ZN0 r 2 s i n e d e d)~ U 3 a -3 x e x p [ - ( U 2 + V2 - 2UV × cos 7 cos¢)/2~ 2] × Jo (iU V sin 7 sin¢/a 2) x sin7 cos (180 ° - 7 ) d U d ~ ? ,
× (1 -}- qz)z -j/2 /KVzl/2\ × sinh [ ~ ) d q d z .
(8)
where Jo(iX) is the Bessel function of the first kind of the order zero. A t this point, we express U and ~ in t e r m s of the orbital elements of interstellar comets: perihelion distance q and semim a j o r axis a. Recalling t h a t U is the velocity u n p e r t u r b e d b y the Sun, and p u t t i n g z = --1/a >~ O, we h a v e (2) U=limK r +z
1/2
= ~ z l/z,
2a 2/
(9)
r~oo
where K = 2 9 . 7 8 (AU)~/2kmsec-L W h e n r - + ~, t h e n for a n y finite q the angles ~b -+ 0 ° a n d 7 -+ 180°, and hence cos~b -+ 1
(12)
Finally, integrating (12) over all perihelion distances from 0 to q and over all negative semimajor axes (i.e., over z from 0 to oo), we h a v e the total influx rate of interstellar comets with perihelion distances less t h a n q:
= f:=o f z:o , ( q , z ) d q d :
=
2,,Uoq F(q), (13)
where
K2 rl V ~ [ V2 + ~ 2 ] F(q) = ~ e r z / - - , ~1 + q
q~
+(2~exp
(- ~ -V2 ) ~ ,
(14)
and erf (x) is the error integral of a r g u m e n t X.
126
ZDENEK SEKANINA
I I I . THEORETICAL DISTRIBUTION OF SEMIMAJOR A X E S OF INTERSTELLAR COMETS
To demonstrate the general limits for systematic velocities V and velocity dispersions a t h a t could a priori be considered in describing the dynamical behavior of a cloud of interstellar comets, we have compiled Fig. 1 from the available d a t a on virtually all classes of stars and other galactic objects as listed mainly by Delhaye (1965) and Allen (1973, pp. 252-254). The m e a n dispersion a must be computed from the dispersion components an, (~o, az in the three f u n d a m e n t a l galactic directions, and, of course, there are various ways to define the function a(an, as, az). Let. us consider two cases. First, one can adopt a "stereometric" approach and require t h a t in the velocity space the " v o l u m e s " of the observed dispersion ellipsoid and of the postulated dispersion sphere be equal, in which case a = (anaoaz) t/3 (see Fig. 1). Or, second, one can adopt a "vect o r - s u m " approach and require t h a t the magnitudes of the two dispersions, in the ellipsoidal and spherical distributions, be i
i
~ iJ
I
~
i
i
[
i
~ J~,l
J
/
i
I ÷-,o-~
200
I
E
:~"
/
/
..o.- ~-.,o
s"°°°//
I00
I I
/
/
/
~. 5o
(
II, / /
/
.o"
~"
.~
.o
.¢.'..o"" :~''t"
p,.
[ // / /I
o
/I
/
I
I
~ l ~l[ 0
5 MEAN
VELOCITY
DISPERSION
F
P
I
50
I
f q ~(q,z)dq dz p.(z)~tz=~_q=o
I [
o/ 1
20
U I
"o"".~':'g':. "o".//~,¢o.,/ - • o
[
I
o.o
I
Jill
I
I00
cr = (CrriO-(jO-z)1/3
equal, in which case a = [½ (an 2 + aa 2 + az2)] 1/2. We note t h a t the latter approach, leading to the quadratic mean, tends to overemphasize the contribution from the m a x i m u m component (which is usually the radial one, an), while the former approach, resulting in the geometric mean, gives more weight to the m i n i m u m component (which is often the normal one, az). To check the effect of nonsphericity of the actual velocity-dispersion distributions, a relative difference between the two definitions of a has been calculated for each entry of Fig. 1. On an average, the geometric mean comes out to be 7 ± 6% smaller t h a n the quadratic mean, which is interpreted as an indication t h a t the simplifying assumption (2) is a fair approximation for our purposes. The approach developed in Section I I is further used to calculate the distribution of semimaj or axes for various combinations of V and a covering the whole populated area in Fig. I, and to draw conclusions as to the degree of likelihood of existence of interstellar comets. The well-known absence of strongly hyperbolic orbits among the observed comets is now complemented by the detection (Marsden and Sekanina, unpublished) of a few slightly hyperbolic orbits whose original 1/a are of order -0.00001 to -0.0001 (AU) -I (for a note on these comets, see Whipple, 1975). The probability t h a t an interstellar comet with perihelion distance smaller t h a n q has an original semimajor axis between z and z ÷ dz is, according to (l 2) and (13),
200
- (K/a) (1 + qz/2) z -1/2 exp (-,:Zz/2a 2) x sinh (KVzl/Z/a 2) × {(2~r)1/2 erf (V/a~/2) exp (]/'2/202) > [1 + q( V2 + 02)/2K z] + qVcr/K2} -1
dz.
(15)
{ k r n sec - I )
Fl(~. 1. M e a n velocity dispersi(m a = (a//aoaz) 1/3 v e r s u s s y s t e m a t i c velocity V relat i v e to t h e Sun for v a r i o u s classes of galactic o b j e c t s (solid circles). O p e n circles are s t a n d a r d c o m b i n a t i o n s of ,s a n d V. for w h i c h c a l c u l a t i o n s h a v e b e e n m a d e (Table I I , Fig. 2). C o m p a r e t h e p l o t t e d l i m i t s for coefficients A a n d B w i t h thos(, applied in T a b l e I.
The q u a n t i t y t h a t is of primary interest, from the statistical point of view is the median value zo = - ( l / a ) c , for which the occurrence of hyperbolic orbits with z > zo equals that, of orbits with z < zc ; i.e.,
f;'pq(z)dZ= fz~pq(z)dz= l.
(16)
E N C O U N T E R S W I T H I:NTERSTELLAR C OME:rS
Expression (15) depends on three parameters, q, V, and a, b u t for the purpose of determining %, we can tie t h e m into two dimensionless coefficients, A = q~Z/2K2, B = V/a~/2, (17) and calculate z~ t h r o u g h ~ from - . :c p ( q ( ) d ( ,
(18)
in which
pq(~) d~ = [1 + (A~2/2)] exp (-~2/4) sinh (B~) d~ 7r1/2 erf (B) exp (B 2) (1 + A + 2 A B : ) + 2 A B (19)
127
and the dimensionless variable ~ is related to z b y z = (A/q) ~2 = (a2/2K2) ~2.
Table I lists ~c for a n u m b e r of values of A and B. The limits selected for B cover all the combinations of V and a in Fig. l, while the range in A is chosen so t h a t , in addition, the perihelion distance can be varied at least between 1 and 3AU. The m e d i a n semimajor axis can now be established for an a r b i t r a r y choice of V and a o f the p o p u l a t e d area of Fig. 1 b y i n t e r p o l a t i n g ~c in Table I and applying (20). F o r q within 1.2AU and several s t a n d a r d combinations
TABLE
I
QUANTITY ~c, ]~EI.ATED TO THE MEDIAN SEMIMAJOR A x I s OF INTERSTEI, LAR COMETS. AS A FUNCTION OF COEFFICIENTS A AND B
A B l 0 -°'7 10 -o.4 10 - ° ' a 1 10 °'1 10 °.2 10 °'a 19 °'4 l0 ° s
10 - ~ ' °
10 -1"5
10 - l ' °
10 - ° ' s
1
10 ° ' s
101"°
101"5
1.70 1.77 1.92 2.31 2.70 3.27 4.07 5.11 6.43
1.74 1.81 1.97 2.38 2.78 3.38 4.20 5.25 6.57
1.84 1.92 2.10 2.56 2.99 3.61 4.43 5.45 6.73
2.06 2.16 2.37 2.86 3.30 3.89 4.66 5.62 6.85
2.34 2.45 2.66 3.13 3.54 4.08 4.78 5.70 6.89
2.52 2.62 2.82 3.27 3.65 4.15 4.83 5.73 6.91
2.59 2.69 2.89 3.32 3.68 4.18 4.85 5.74 6.92
2.61 2.71 2.92 3.34 3.70 4.19 4.86 5.75 6.92
TABLE
II
MEDIAN I~ECIrROCAL SEMIMAJOR AXIS (1/a)c FOR STANDARD COMBINATIONS OF SYSTEMATIC VELOCITY W AND DlSVERSION a (q ~< 1 . 2 A U ) V (kmsec -l)
a ( k m s e c -1)
10 10 15 15 15 20 20 20 20 30
l0 30 7 15 25 5 10 20 40 15
(l/a)c ( A U ) -1 --0.252 --2.55 --0.292 --0.650 --1.79 --0.475 --0.573 --1.31 --5.34 --1.46
(20)
V ( k m s e c -1) 30 30 40 50 50 70 70 100 175 300
a ( k m s e c -1) 30 50 40 30 70 40 60 75 100 125
(1/a)¢ ( A U ) -1 --3.51 --9.25 --6.86 --5.29 --20.2 --ll.1 --18.5 --32.6 --74.8 --168
] 2N
ZDENEK SEKANINA 7
7
----7
•
[
(10,10)
,/"
(20.5) / (!5,7) (15351
(20,10) 130, t5) "'
(20
oO1
'=
/
(50,30)
0q(l/o) (iU)
(40,40)
o~
i
OOl
I
1_ . 2
i
.
.
.
.
.
.
ORIGINAL RECIPROCAL SEMIMAJOR AXIS,
Fl(i. within ~ i i l i l" ]lo}ioll })q(lia)
1 ~
I/o
4
(AU) ,I
2. l ) i s t r i l ) u t i o n tJ,~(lla) (d" r(,cil)roclit s(,l/liiiillj(>l > llx(,s (~f Jnt(~rst('llar (~oin('ts w i t h p e r i l i e l i a l . g A U f]'oni i, ho ~illi. T h ( ' t w o i~]711l'(~s lit cacti Olli'V(~ I2i\:(' tllo s y s t o n m t i e v o l ( ) e i i y relat i\'e t() l lie a n d t i m v o h ) c i t y disi)(,rsi(tn cs (t)()tli hi l~:nls(,o - i ), r('sl)(~ctiv('l.v. T i l e fi'a(;{iOli o f t.onlet>~ w i t h p(,ri(lista, n(,es q % 1 . 2 A U t l l a t h a v e r~!cipro('fll s(~ininlaj()r a x e s I)(,tween ]la a n d l l a + d ( l / ' a ) is
d( l /a).
o{' 1" and ~, which are marke<| in Fig. I by open circles, the median (l/a)c is listed in T a b l e I I . The
(21)
( ; = A ÷ ~B 2.
If (i <: 1, the curve decreases from the beginning: if(;== ~,1 it a t t a i n s a l n a x i m u m a t I l a = (): and if'(/:> 2,its m a x i n i u m t a k e s place at (l/a),,, z~ < o, which relates to ~,,, according to (20), where ~,, satisfies the e(luation t a n h (B~,.) B~,,l l + (A~.//:)I
I I + (~,,,->/2)1 l1 + ( A ~ , f - / 2 ) [
. ~4~,.:
(22) E v e r y piece of informat.ion gained from s t u d y i n g the theoretical distribution of s e m i m a j o r axes of interstellar comets points to the s a m e conclusion: ()n the ass u m p t i o n of a similarity in the dynanficaI b e h a v i o r between the eloud of these comets a n d one of the k n o w n classes of galactie objeets, the <)rbits of the o v e r w h e h n i n g m a j o r i t y of interstellar comets, if we saw
t h e m , would be st,rongly hyperbolic, with I / a *~ = 0. I ( A U ) - ' ; i.e., t h e y would d e v i a t e ti'om a p a r a b o l a b v at ]cast 3 orders of magnitude more a p p r e c i a b l y t h a n does the most hyperbolic c o m e t a r y orbit aetually observed. The results obtained for q # 1.2 AU show only a s l i g h t - t o - m o d e r a t e depen
The a p p a r e n t absence of interstellar comets a m o n g tile o b s e r v e d comets can be used, following Eq. (13), to establish an upper limit on the space c o n c e n t r a t i o n of interstellar comets in tile solar neighborhood. In order to have figures readily coral)arable with those available on other classes of objects in the G a l a x y , it is useful to convert the space concentration N 0 into the space density P0 of interstellar comets. E x p r e s s i n g the latter in solar masses per cubic parsee, we have, fi'om (13), Po = g l ( q ) / c q
F(q),
(23)
where il~(q) is the actual mass influx, in g r a m s per century, of interstellar comets with periheli<)n less t h a n q, c :- 3 ,~, ] ¢p9, q
ENCOU:NTERS
WITH
INTERSTELLAR
is in AU, a n d F(q) is in A U k m s e c - l . To express P0 in g c m -3, its value in MGpc -3 m u s t be multiplied b y a f a c t o r of 6.77 x 10-23. To establish an u p p e r limit on P0, we det e r m i n e first the c h a r a c t e r of the dependence of expression (23) on t h e s y s t e m a t i c v e l o c i t y V relative to the Sun a n d on t h e v e l o c i t y dispersion a. I t can easily be shown t h a t p e a k space densities of interstellar comets for an a r b i t r a r y choice of M(q) a n d q are located along a c u r v i n g " r i d g e , " a t t a i n i n g the m a x i m u m v a l u e of Pmax =
~f(q)
(~/2c/¢q3/2)
-1
(24)
a t the p o i n t V = Vo = K(2/q) 1/2, a ~ O, a n d slowly sloping d o w n to P'0 = 0r/2) 1/2 3;/(q) (2cKq3/2) -1 = (rrl/2/2)Pmax
(25)
at the p o i n t a = a o = Kq - 1 / 2 : VolVO-2, V - + 0. The d e n s i t y d r o p s r a t h e r steeply on b o t h sides of the ridge, b u t g r a d u a l l y levels off a t V >> V0 a n d ¢ >> %. The actual m a s s influx M can be determ i n e d as the m a s s influx of discovered comet.s p divided b y their m e a n discovery p r o b a b i l i t y G. B o t h q u a n t i t i e s d e p e n d not only on perihelion distance, b u t also on the intrinsic brightness I 0. F u r t h e r m o r e , the influx t~ is a p r o d u c t of the discovery r a t e 8 (comets per century) a n d the m e a n m a s s of the discovered comets_M. Thus, in total, ;if(q) =: 3(q, AIo) M(q, AIo)[G(q, A / o ) ] - ' , (26) where q indicates the m a x i m u m perihelion distance considered, while A I o s t a n d s for the considered r a n g e of the l u m i n o s i t y distribution. F i r s t of all, as a p p a r e n t l y no interstellar c o m e t was discovered during the p a s t 100 y e a r s or so (since which time, fairly precise orbits h a v e been available for p r a c t i c a l l y e v e r y o b s e r v e d comet), we adopt, as a v e r y c o n s e r v a t i v e u p p e r limit, 8 < 1 interstellar c o m e t per c e n t u r y regardless of q a n d A I o. N e x t , to assess M, we a s s u m e t h a t the d i s t r i b u t i o n of c o m e t masses is g o v e r n e d b y a p o w e r law M -~ d M , where Mml n <
] 29
COMETS
M < Mma x a n d s is a constant. F o r Mma x >> Mmin, the m e a n c o m e t m a s s implied is :
[(s
1)/(2
__
s)]
2 -as x M msi - ~n 2~I m
(s # l, 2).
(27)
Several a t t e m p t s were m a d e r e c e n t l y to d e t e r m i n e the e x p o n e n t s or a related p a r a m e t e r . E v e r h a r t ' s (1967) results indicate t h a t the intrinsic d i s t r i b u t i o n of longperiod comets varies as l o 5/2 d I o for comets b r i g h t e r t h a n absolute m a g n i t u d e 5.3 and as 1o s/3 d1 o for f a i n t e r ones. W i t h the massl u m i n o s i t y relation p r o p o s e d b y W h i p p l e (1975), these laws give s = 2 a n d s = 13/9, respectively. Using " n u c l e a r " m a g n i t u d e s of comets, R o e m e r (1968) f o u n d that. the c u m u l a t i v e n u m b e r of comets varies in inverse p r o p o r t i o n to the 1.5 p o w e r of the nuclear size, which corresponds to s = ~. F r o m V s e k h s v y a t s k y ' s catalog of t o t a l absolute m a g n i t u d e s , W h i p p l e (!.975) obt a i n e d a law l o 2"1 dlo, s = 1.73, for n e a r l y parabolic comets w i t h q < 1.2 A U o b s e r v e d during the period 1850-1949, e s t i m a t i n g t h a t the m a x i m u m m a s s for a c o m e t a r y b o d y is a b o u t Mma x ~ 1022g. The m i n i m u m m a s s Mini n is d e t e r m i n e d b y a limit imposed on the discovery p r o b a b i l i t y t;. E v e r h a r t (1967, T a b l e I I ) lists G as a function of perihelion distance a n d absolute m a g n i t u d e . F o r q < 1.2AU, G does not d r o p below a b o u t 0.2 for comets b r i g h t e r t h a n absolute m a g n i t u d e about. 7-7.5, for which W h i p p l e ' s m a s s - l u m i n o s i t y relation gives Mmi n ~ 1015g. Thus, we find f r o m (27) t h a t the m e a n m a s s M in our case is 10~S'Sg w i t h R o e m e r ' s d i s t r i b u t i o n law; l 017.3 g w i t h W h i p p l e ' s ; and, from a modified form of (27), 1016"6g with E v e r h a r t ' s . A t the s a m e t i m e we find from L v e r h a r t s T a b l e I I t h a t ~7 > 0 . 3 for these comets. I n s e r t i n g into (26) the respective limits for $_and G, plus the g e o m e t r i c m e a n of the t h r e e M ' s , we get a reasonable u p p e r limit on the actual m a s s influx of interstellar comets : 3)/(1.2) < 1 0 1 S g p e r c e n t u r y .
(28)
W e note t h a t the limits o f q a n d I 0 h a v e been t a k e n r a t h e r arbitrarily. W e a d h e r e d to the limit on q picked u p b y Whipple, which allowed us to use some of his results w i t h o u t introducing a n y corrections. W e
I
3(i)
ZDENEK S E K A N ] N A
p o i n t o u t , h o w e v e r , that, w e w o u l d n o t g a i n a n y t h i n g f r o m s t r e t c h i n g q to m u c h larger values, b e c a u s e o f t h e r a p i d l y (leereasing d i s c o v e r y p r o b a b i l i t y at. large q ( E v e r h a r t , 1967). Similarly, e x t e n d i n g the lum inosity d i s t r i b u t i o n t o w a r d f a i n t e r c o m e t s woul(t n o t result in a b e t t e r v a l u e o f 3.)r, p a r t l y again, b e c a u s e o f a low d i s c o v e r y proba.bilitv o f faint comets, ~md p a r t l y also because o f t h e s u b s t a n t i a l increase in the unc e r t a i n t y e l M due to t h e error in t h e exp o n e n t s. A d o p t i n g 3:~ f r o m (2S) an(t
fbr l" ~ 0, ~ ~ 0, this limit r e m a i n s on t h e o r d e r o f m a g n i t u d e o f a few t i m e s 10 -4 M pc -~ for ~dl w d u e s of" I" a n d (~ t h a t could possibly be considered. These figures confirm %Thipple's (1975) r e c e n t result d e r i v e d f r o m a simplified t h e o r y . H o w e v e r , since we h a v e been v e r y c o n s e r v a t i v e in c h o o s i n g the n u m e r i c a l v a l u e s for t h e q u a n t i t i e s s e t t i n g p . . . . a m o r e realistic u p p e r limit m i g h t well be an o r d e r o f m a g n i t u d e l o w e r t h a n (29), an(t, o f course, t h e a,c t u a l spa(:(, (lensity o f interstellar c o m e t s may. still [)e m a n v orders o f m a g n i t u d e h)wer t h a n t h a t . In t e r m s o f t h e space c o n c e n t r a t i o n , t h e m a x i m u n l d e n s i t y w o u l d i n d i e a t e the ln'esenee of" o11(, eolnet p e r si)her(, 1 2 A [ r in ra,di
V. A CLOUD OF [NTERSTEI,LAR (IOMETS T R A V E L I N G WITH THE S[:N
Pmax <: (J X 10 -4 M ,l)e -3 : : 4 ::< I 0 -'¢'Kcm --3.
(2:)) or a b o u t ( ) . 5 % o f t h e t o t a l d e n s i t y o f t h e m a t t e r in the solar n e i g h b o r h o o d . E x c e p t
I
j
us.
T h e case o f zero s y s t e m a t i c v e l o c i t y o f a cloud relative to the S u n h a s n o t been covere(1 explicitly in t h e p r e v i o u s sections. T h e general t h e o r y f o r m u l a t e d in Sections
UPPER LIMIT ON i (10-4M° PC-5)
____
/ ::.\
\
/ i
20
/a we
i
'0
20
40
60
80
I00
G (kin sec-I)
FH'. 3. Upper limit on the spa,ee density ofinter.~tellm'conu'ts with pcriheliawithin 1.2AU a~ a flmction of th(' velocity dispersion cr and th(, system~ti(- \eloeity 1" relative to the Sun. Not.(, tht. m~ximum al cr=(L 1"=38.4kms~'(: ~. and th(" "ri(bze" M~)ping (l()wn t() or= 27.2kms('( '-I. I ' = 0 . whm',, tim limit on the sl)a(.~' d(,~J~ily ('¢lual 0.8.q ,)f'lh(, ]D~IXilIIIIIIIx.'IINHL
ENCOUNTERS
WITH INTERSTELLAR
I I and I I I gives the following relations for V-~0. /¢2 (
p~(z)dz=-~o2
and, for q .> K2/O"z or A/>- 1,
z,, = 2(az/• z - l/q)
O.2) --1
l+q~
× exp ( - h
(34) Our calculations for interstellar comets traveling with the Sun are s u m m a r i z e d in Table I I I . Unlike in Section I I I , we now would e x p e c t most interstellar comets to have nearly parabolic orbits, as long as the velocity dispersion in the comet cloud is k e p t low, well u n d e r l k m s e c -1. The space density implied is t h e n of order 10-SM+ pc -3 or less ( < 1 0 - 2 7 g c m - 3 ) . The present formulation of the problem (neglecting solar a t t r a c t i o n on the comets) does not provide for a capture-prone or a S u n - b o u n d comet cloud even in the case V = 0, a -~ 0. H o w e v e r , because of the low S u n - c o m e t velocities involved, our idealized approach is no longer adequate, and this case becomes a problem of an 0 o r t - t y p e cloud, where solar and stellar p e r t u r b a t i o n s m u s t be t a k e n into account. Unless ~ - + 0, however, Table I I I indicates t h a t strongly hyperbolic orbits result overwhelmingly, even in the case of zero systematic velocity V, and t h a t the u p p e r limit on the space density of interstellar comets is t h e n practically identical with the estimates derived from the general t h e o r y (Section IV).
lqz)
z) dz,
(30)
(+-)
× exp - ~ z
= 1 - Ill ÷ 1 (1 + 2A) -~qz] × exp ( - h z )
PO = ~
,
(31)
/¢2 ÷ qo.'-'''~
_ ~r(q) (~A) '/~ (1 + 2 A ) - ' , cscq 3/2 {2
-2Alog¢{211+a(1 7
= (2/q) (2A - 1).
(l + ½qz)
q (1 + 2A) -~ (1 + 4A
2cr2
] 31
COMETS
(1
(32)
O'2~--1
+ 2 A ) -~qzc] },
(33)
TABLE
11I
( I / a ) DISTRIBUTION PARA=~IETERS AND SPACE--DENSITY ESTIMATES lrOR INTERSTELLAR COMETS (q ~ 1 . 2 A U ) TRAVELrX(' ~VITH THE SUN
(W : 0 ; SOLAR ATTRACTION NEGI,ECTED)
( 1/a)~
( 1/a ),.
( k i n s e e -1 )
(AU) -l
(AU) -l
0.1 0.2 ().5 1 2 5 ]0
--0.0000156 --0.0000625 --0.000391 --0.00157 --0.00629 --0.0404 --0.176
--------
~10 ~10 <10 <10 <10
--
<10 -3.7
<10 -26.3 < 1 0 -25,9
--
< 10 -3"4
< 1 tl - 2 5 . 6
21)
--0.895
--
< 1 0 -3.3
<10 -25.5
--0.363 --3.97 --9.38
<10 -3.3 < 10 - 3 . 3 <10 -3.4
<10 -25.5 < 10 -2s-5 <10 -25.6
30 50 70
--2.43 --8.12 --17.1
po (M+pc -z ) -4.s -4.s -4.7 -4.4 -4"1
(gem -3 ) ~10 -27.°
~ 1 0 -26"7 < 1 0 -26.9 <10 -26.6
132
ZDENEK SEKANINA V[. (:(>N(JLUSIONS
()bserw~tional evidence virtually rules out the existence, in the solar neighhorhood, of a cloud of interstellar comets with a s])ace density of 6 × l 0 -4 solar masses l)er cuhic parsec ( 4 x lo-><~gcm 3) or higher and makes the existence of a conlet cloud l(i times less dense v e r y unlikely. However, the probability of e n c o u n t e r with the Sun of a comet from a cloud whose space density does not exceed l0 ~ solar masses per cubic parsec (hess than 10 e7gcm 3) is too low to exchlde a priori the, existence of interstellar comets~dtogether. For comparison, we note t h a t O o r t ' s (1950) estimates of the size a n d p o p u l a t i o n of his model of a solar-bound comet cloud give a nlean sl)ace (lensity of only a b o u t l0 -6 solar masses per cubic parsec (~10-_>8gcm 3), although Whipple (1975) has recently shown t h a t the possible unobserved mass of comets in the solar system could be a not-insignificant fra(..tion of the solar mass. The hypothetical interstellar comets might have originated in. s i l u o r might have be(m, at birth, s t a r - b o u n d comets t h a t were later ejected from circumstellar areas in a m a n n e r similar, perhaps, to the expulsion of a fl'action of nearly parabolic comets fronl the solar system b v p l a n e t a r y lierturhations. I n e i t h e r case, the m o t i o n s o f such interstellar colnets should sh(m an a|)liF('ciable s y s t e m a t i c velocity relative to the Sllli (II1OI'(. ~ t h a n l()kmsee
i) an(t a (lisper-
sion in in(lividual veh)cities of at least 5knlse( :--~, judging fl'oln analogy with other groups of galactic objects. While a (listinct systenlatic motion of the cloud relative to the Sun implies the Iiresence of a iweferential direction in the distribution of eonletary aphelia, which was detected I)y some (e.g., Tyror, 1957 ; H urnik, 195(.)) in the motions of nearly l)ar~boli(: comets, it also re(tuires t h a t an overal)un(tance o f s t r o n g l y h y p e r b o l i c o r b i t s be observed, w h i c h , o f course, has never materialize(I. ()n the o t h e r hart(l, the a p p a r e n t al)sen(;e
()fstrongly hyl)erbolic orl)its, together wilh the scarce occtlrrence o f s l i g h t l y hvI)er
holh: orbits alnollg the ohserve(l ('Olllets~ c()uhl only ilnl)ly a clou(l of comets travelilig w i t h the Sun (no s y s t e n l a t i e velocity
and a very low dispersion), but the aplielion l)oints would then have to be distribute(t essentially isotropically. Since an alternative and v e r y plausible explanation ( n o n g r a v i t a t i o n a l effects) is at hand for the few slightly hyperbolic original orbits t h a i are observed, we do not find it, a t t r a c t i v e to link a n y of the orbital properties of the actual comets with the hypothesis of interstellar comets. Tim model considered here coul(I, of course, lie generalized even further by considering an ellipsoidal distribution of coinet velocities (11, i.e., by w i t h d r a w i n g restriction (2). In the light of our present results, howeve, r, such a. move suhstant i ally co in plicati ng the m at he in ati cal treatment of the p r o b l e m - - d o e s not a p p e a r to be worth the effort, not at least in the fi)reseeable future. A('KNOWI,EDGMENT T h i s r(,search ~ a s mipport(,(| I)y ~_,rant N8(~ 7t)b12 fr()lii t h e Nat:toned A e r o n a u t i c s ~liid Sioaot~ Adlniliis(,rlt I i()ll. |{EFERENCES AI,I,I+]N, (I. li~'. (19731.
A,vtrophy,s'ical Qna~.titie,~'.
3rd od. Alhhmo Press, London.
Db:LttAVF, J. (19651. Solar mot,ion and velocit.v distributior~ of (.mm.m stars. In Ualactie Ntr,ct,,re (A. I~hlaux~ and M. Selimidt, Fds.), pp. t;l ~44. University of (',hiealzo lh'0ss, Chiea~'o. EVEltHAI~T, E. (19(;7). Intrinsic distrilmlions ~d' eolnctary poriholia and nla~nitmh's. , t Mro/#.,]. 72. 1002 1011. HVI:BSFI~, 1,'V. F., SN,, I)ER, 1,. E., AND I'll:Ill,, D. (1974). HCN r0x|io (,nlission frolll (~oiiilq I{oltoutol-: (1973f). learu.s '~3, 580 584. iluI~NIK, H. (19591. The distmibution of the directions of porihcdia, an(I of the orbital p, dos ( )f' n o i i .| )l q.il )d i(~ f~o n v.,l s, .q cla.4,~;t~"olt,. 9, 2t ) 7 ")_21.
?tlARSI)I.ZN. B. (I-., ANI) ~EKANINA. Z. f l 9 7 3 ) . ()it th(' d i s t r i b u t i o n o f "ori~ziiml" ()l'])ItS
Ills
1124.
~'IAI'~SliEN. I/. (L. ~I.]KANINA, Z., AND YEOMANS.
1), K. (19731. (loinets mid ll.nffravitatiollal f'or('cs V. Astro~l.J.78, 211 225. ()owr..I. 11. (19501. The st,ruct,uro of the clolld of (,I)lil('ls s l l r r o t i n d i l i ~ t h e solar s y s l o i n , a n d it [i3'l)l)lh(~sis (!OilOtWllili 7" it, s o r i g i n . 11.11. A.s'tr
ENCOUNTERS WITH INTERSTELLAR COMETS RICHTER, N. B. (1963). T h e N a t u r e of Comets. Methuen, London. ROEI~fER, E. (1968). Dimensions of the nuclei of periodic and near-parabolic cornets (abstract). Astron. J . 73, $33. TYROR, J. C~. (1957). The distribution of the directions of perihelia of long-period comets. M o n . Not. Roy. Astron. Soc. 117, 370-379.
] 33
ULICH, B. L., AND CONKLIN, E. K. (1974). Detection of methyl cyanide in Comet Kohoutek. N a t u r e 248, 121-122.
WHIPPLE, F. L. (1975). Do comets play a role in galactic chemistry and y-ray bursts? A stron. J . 80, 525 531.
A Probability of Encounter w i t h Interstellar C o m e t s and the Likelihood of their Existence ZDENEK SEKANINA Center for Astrophysics, Harvard College Observatory and Smithsonian Astrophysical Observatory, Cambridge, Massachusetts 02138 l~eceived May l, 1975; r e v i s e d J u n e 26, 1975 A t h e o r y o f t h e p r o b a b i l i t y o f e n c o u n t e r o f t h e Sun w i t h an i n t e r s t e l l a r c o m e t a t a d i s t a n c e c o m p a r a b l e to t h e E a r t h - S t m d i s t a n c e is f o r m u l a t e d , a n d a general e x p r e s s i o n is d e r i v e d e s t a b l i s h i n g t h e r e l a t i o n s h i p a m o n g t h e influx r a t e o f interstellar c o m e t s , t h e perihelion d i s t a n c e , t h e space d e n s i t y o f t h e c o m e t s , t h e M a x w e l l i a n d i s t r i b u t i o n o f c o m e t velocities in t h e i n t e r s t e l l a r cloud, a n d t h e c l o u d ' s s y s t e m a t i c v e l o c i t y relative t o t h e Sun. The f a c t t h a t no c o m e t w i t h a s t r o n g l y h y p e r b o l i c o r b i t h a s so far b e e n o b s e r v e d is u s e d t o d e t e r m i n e a n u p p e r limit o f 6 × l 0 -4 solar m a s s e s p e r cubic parsec (4 × 1 0 - 2 6 g c m -3) for t h e space density of interstellar comets. The theoretical distribution of semimajor axes of i n t e r s t e l l a r c o m e t s is d e r i v e d t o s h o w t h a t a s t r o n g h y p e r b o l i c excess m u s t be p r e s e n t in t h e o r b i t s o f a m a j o r i t y of i n t e r s t e l l a r c o m e t s regardless of t h e d y n a m i c a l c h a r a c t e r i s t i c s o f t h e c o m e t cloud, e x c e p t w h e n t h e cloud is m o v i n g along w i t h t h e Sun a n d t h e d i s t r i b u t i o n o f i n d i v i d u a l velocities has a v e r y low dispersion. This case, h o w e v e r , implies a p o s s i b i l i t y o f c a p t u r e b y t h e Sun a n d t h u s b e c o m e s a p r o b l e m o f a n O o r t - t y p e cloud. I . INTRODUCTION
The strong dynamical evidence corroborating Oort's (1950) hypothesis of a cometary cloud permanently associated with the solar system has recently been further reinforced by Marsden et al. (1973), who show t h a t the proper account of the nongravitational effects in the motions of nearly parabolic comets leads to smaller dimensions of the Oort cloud and therefore to its stronger gravitational bonds with the solar system, and by Marsden and Sekanina (1973), who find t h a t the aphelion distances of nearly parabolic comets least affected by the nongravitational forces pile around 50 000AU. On the other hand, some of the recent accomplishments in the physics of comets - - i n particular, the discovery in Comet Kohoutek (1973f) of methyl cyanide (Ulich and Conklin, 1974) and hydrogen cyanide (Huebner et al., 1974), two species previously detected in interstellar space-suggest t h a t more attention should be paid in the future to the relationship between Copyright © 1976by AcademicPress,Inc. All rights of reproduction in any form reserved. Printed in Great Britain
comets and interstellar matter. Although the discovery of interstellar molecules in comets does not offer any straightforward interpretation in terms of the interaction between comets and interstellar matter, it may, at least temporarily, place the theories of interstellar comet origin into a more favorable light, especially when assisted by some previously known circumstantial evidence [such as the anisotropic distribution of comet-aphelion directions; see, e.g., Tyror (1957) and Hurnik (1959)]. The aim of this paper is to investigate the possibility of the existence of a cloud of interstellar comets in the Sun's neighborhood. The study is based on a new approach to the relationship between the space density of interstellar comets and their distribution of velocities in the cloud, on the one hand, and their rate of influx to the observable region of the solar system and the distribution of their orbital elements on the other. Some work along these lines was done in the past [for a brief review of the research up to about a decade ago, see Richter (1963, pp. 147-152)], hut
123
124
ZDENEK S E K A N I N A
the results seem to d e p e n d r a t h e r strongly on the starting assumptions. The present a p p r o a c h is f o r m u l a t e d with a m i n i m u m of necessary restrictions; y e t it is mathematically t r a c t a b l e with relative ease. II. GENERAL EXPRESSION FOR THE INFLUX R A T E OF INTERSTELLAR COMETS
As in a n y other class of objects in the Galaxy, the members of a cloud of interstellar comets m u s t have a certain velocity distribution relative to their center of mass. I f the gravitational interactions a m o n g the individual objects of the class a n d the p e r t u r b a t i o n s from the Sun and o t h e r a m b i e n t objects are neglected, the stellar statistics show t h a t the normalized velocity distribution within a n y such class of objects can be a p p r o x i m a t e d b y the Gaussian distribution function ¢ ( v x, vy, v~) dv x dv r dv z = (27r) -3/2 (axCry(yz)-1
i_[fVx: r_x)2
× exp |
+
[
2ax2
+
v )2
2at2
( V z - V~)2]i
]l VxdV dV ,
where the directions of the three axes of the reference system of coordinates are det e r m i n e d b y the vectors x, y, z. The velocity distribution is defined b y six constants: Vx, Vr, V~ are the c o m p o n e n t s of the velocity o f the cloud's center o f mass relative to the Sun, and a~, %, a~ are the c o m p o n e n t s of the s t a n d a r d velocity dispersion in the cloud. Then, ¢(vx, vr, Vz) dvx dvy dv~ is numerically the relative fraction (percentage) of comets in the cloud, for which the velocity components relative to the Sun lie between v~ and vx ÷ dvx, vr and vr + dvy, and v~ and Vz + dvz. To reduce the n u m b e r of u n k n o w n s and to facilitate the integration substantially, we assume a spherical r a t h e r t h a n an ellipsoidal distribution of velocities; i.e., we define ax = % = a~ - ~.
(2)
This assumption also allows us to identify, w i t h o u t imposing a n y f u r t h e r restrictions, the direction of m o t i o n of the cloud's center
of mass with a n y one of the three coordinate axes. W e therefore define a new system of coordinates (x, y, z), whose center is located in the Sun and whose +z axis directs along the v e c t o r Y of the systematic velocity of the center of mass of the comet cloud relative to the Sun. I f U is the vector of an individual relative velocity referred to the Sun, and u, v, w are its c o m p o n e n t s in the three axes, distribution (1) becomes ~5(u, v, w) du dv dw = (2~r)-3/2 ~-3 × exp [ - ( U 2 + V 2 _ 2Vw)/2a 2] × du dv dw.
(a)
We recall t h a t ~5 is the distribution o f c o m e t a r y velocities before a n y appreciable gravitational pull of the Sun has commenced. N e x t we consider an e l e m e n t a r y surface area d S located at (xo, Yo, %) on a sphere o f radius r centered on the Sun. I f N o is the spatial concentration o f comets (per unit volume), assumed to be constant throughout the cloud, and is(U, v, w, Xo, Yo, %) is the flux rate of comets at d S (per unit area per unit time) at velocity U = (u, v, w), t h e n the n u m b e r of interstellar comets t h a t pass t h r o u g h d S per unit time with velocities between U and U + d U and with velocity c o m p o n e n t s between u and u + du, v and v + dv, and w and w + dw, respectively, is is(U, v, w, x o, Yo, Zo) cos ~bd S du dv dw = No r2 U ~(u, v, w) × cos~bsin¢d$dAdudvdw,
(4)
where ~b is the angle between the velocity vector U and the direction from d S t o w a r d the Sun, ¢ is given b y (3), and the angles and A are the polar coordinates of d S defined b y (Xo)[rsinqbc°sh\ Yo = | r s i n ~ b s i n A ] . zo \ r cos~b ]
(5)
N e x t we t r a n s f o r m the velocity components u, v, w to a n o t h e r system of coordinates, which has its center in the elem e n t a r y area dS, its + Z axis directed along the prolonged radius v e c t o r a w a y from the Sun, and its X and Y axes t a n g e n t to the
125
ENCOUNTERS WITH INTERSTELLAR COMETS
sphere of radius r. F u r t h e r , ~ is the angle from the X axis a r b i t r a r i l y oriented in the t a n g e n t plane, and 7 is the angle c o u n t e d from the + Z axis. The velocity c o m p o n e n t s u, v, w are now related to those of the new system as follows.
(:)
co i)(i 0
0/
=|--COS)t sin2 cos¢ sincb w 0 - s i n ¢ cos ~b/ [ U sin 7 e o s ~ x | V sin 7 s i n $ } , (6)
and cos 7 -->-1. T h e p r o d u c t r sin 7 converges to r sin ~ = lira q
+ z
+ z
r-~ oo
= qz - m
+z
,
(10)
so t h a t
-~z ~ U r ~°71 = l i m K - ~ 2 +
(~
\ Ucos~ / ×
and au Ov aw aU aU OU au av aw
Ou Ov 0~1 O7
-bz
q'-
U z sin 7.
=~--(]+qz) 2qz
(7)
3w 07
This t r a n s f o r m a t i o n allows us to i n t e g r a t e (4) over all possible angles $ and to write the influx o f the interstellar comets with velocities between U and U-4-dU and in directions between 7 and 7 + d~ (90 ° < 7 -<180 °) t h r o u g h dS per unit time:
+z
(11)
I f we integrate (8) over the whole sphere and allow r -~ ¢c, the influx rate (per unit time) o f interstellar comets with original perihelion distances between q and q + dq and original semimajor axes between z and z + dz becomes
i,(q, z) dq dz = ~2~)1/2 No K3 (Va) -~ × exp (-- V:
i~s(U, 7, Xo, Yo, %) cos ¢ dS d U d 7 = (27r)-l/ZN0 r 2 s i n e d e d)~ U 3 a -3 x e x p [ - ( U 2 + V2 - 2UV × cos 7 cos¢)/2~ 2] × Jo (iU V sin 7 sin¢/a 2) x sin7 cos (180 ° - 7 ) d U d ~ ? ,
× (1 -}- qz)z -j/2 /KVzl/2\ × sinh [ ~ ) d q d z .
(8)
where Jo(iX) is the Bessel function of the first kind of the order zero. A t this point, we express U and ~ in t e r m s of the orbital elements of interstellar comets: perihelion distance q and semim a j o r axis a. Recalling t h a t U is the velocity u n p e r t u r b e d b y the Sun, and p u t t i n g z = --1/a >~ O, we h a v e (2) U=limK r +z
1/2
= ~ z l/z,
2a 2/
(9)
r~oo
where K = 2 9 . 7 8 (AU)~/2kmsec-L W h e n r - + ~, t h e n for a n y finite q the angles ~b -+ 0 ° a n d 7 -+ 180°, and hence cos~b -+ 1
(12)
Finally, integrating (12) over all perihelion distances from 0 to q and over all negative semimajor axes (i.e., over z from 0 to oo), we h a v e the total influx rate of interstellar comets with perihelion distances less t h a n q:
= f:=o f z:o , ( q , z ) d q d :
=
2,,Uoq F(q), (13)
where
K2 rl V ~ [ V2 + ~ 2 ] F(q) = ~ e r z / - - , ~1 + q
q~
+(2~exp
(- ~ -V2 ) ~ ,
(14)
and erf (x) is the error integral of a r g u m e n t X.
126
ZDENEK SEKANINA
I I I . THEORETICAL DISTRIBUTION OF SEMIMAJOR A X E S OF INTERSTELLAR COMETS
To demonstrate the general limits for systematic velocities V and velocity dispersions a t h a t could a priori be considered in describing the dynamical behavior of a cloud of interstellar comets, we have compiled Fig. 1 from the available d a t a on virtually all classes of stars and other galactic objects as listed mainly by Delhaye (1965) and Allen (1973, pp. 252-254). The m e a n dispersion a must be computed from the dispersion components an, (~o, az in the three f u n d a m e n t a l galactic directions, and, of course, there are various ways to define the function a(an, as, az). Let. us consider two cases. First, one can adopt a "stereometric" approach and require t h a t in the velocity space the " v o l u m e s " of the observed dispersion ellipsoid and of the postulated dispersion sphere be equal, in which case a = (anaoaz) t/3 (see Fig. 1). Or, second, one can adopt a "vect o r - s u m " approach and require t h a t the magnitudes of the two dispersions, in the ellipsoidal and spherical distributions, be i
i
~ iJ
I
~
i
i
[
i
~ J~,l
J
/
i
I ÷-,o-~
200
I
E
:~"
/
/
..o.- ~-.,o
s"°°°//
I00
I I
/
/
/
~. 5o
(
II, / /
/
.o"
~"
.~
.o
.¢.'..o"" :~''t"
p,.
[ // / /I
o
/I
/
I
I
~ l ~l[ 0
5 MEAN
VELOCITY
DISPERSION
F
P
I
50
I
f q ~(q,z)dq dz p.(z)~tz=~_q=o
I [
o/ 1
20
U I
"o"".~':'g':. "o".//~,¢o.,/ - • o
[
I
o.o
I
Jill
I
I00
cr = (CrriO-(jO-z)1/3
equal, in which case a = [½ (an 2 + aa 2 + az2)] 1/2. We note t h a t the latter approach, leading to the quadratic mean, tends to overemphasize the contribution from the m a x i m u m component (which is usually the radial one, an), while the former approach, resulting in the geometric mean, gives more weight to the m i n i m u m component (which is often the normal one, az). To check the effect of nonsphericity of the actual velocity-dispersion distributions, a relative difference between the two definitions of a has been calculated for each entry of Fig. 1. On an average, the geometric mean comes out to be 7 ± 6% smaller t h a n the quadratic mean, which is interpreted as an indication t h a t the simplifying assumption (2) is a fair approximation for our purposes. The approach developed in Section I I is further used to calculate the distribution of semimaj or axes for various combinations of V and a covering the whole populated area in Fig. I, and to draw conclusions as to the degree of likelihood of existence of interstellar comets. The well-known absence of strongly hyperbolic orbits among the observed comets is now complemented by the detection (Marsden and Sekanina, unpublished) of a few slightly hyperbolic orbits whose original 1/a are of order -0.00001 to -0.0001 (AU) -I (for a note on these comets, see Whipple, 1975). The probability t h a t an interstellar comet with perihelion distance smaller t h a n q has an original semimajor axis between z and z ÷ dz is, according to (l 2) and (13),
200
- (K/a) (1 + qz/2) z -1/2 exp (-,:Zz/2a 2) x sinh (KVzl/Z/a 2) × {(2~r)1/2 erf (V/a~/2) exp (]/'2/202) > [1 + q( V2 + 02)/2K z] + qVcr/K2} -1
dz.
(15)
{ k r n sec - I )
Fl(~. 1. M e a n velocity dispersi(m a = (a//aoaz) 1/3 v e r s u s s y s t e m a t i c velocity V relat i v e to t h e Sun for v a r i o u s classes of galactic o b j e c t s (solid circles). O p e n circles are s t a n d a r d c o m b i n a t i o n s of ,s a n d V. for w h i c h c a l c u l a t i o n s h a v e b e e n m a d e (Table I I , Fig. 2). C o m p a r e t h e p l o t t e d l i m i t s for coefficients A a n d B w i t h thos(, applied in T a b l e I.
The q u a n t i t y t h a t is of primary interest, from the statistical point of view is the median value zo = - ( l / a ) c , for which the occurrence of hyperbolic orbits with z > zo equals that, of orbits with z < zc ; i.e.,
f;'pq(z)dZ= fz~pq(z)dz= l.
(16)
E N C O U N T E R S W I T H I:NTERSTELLAR C OME:rS
Expression (15) depends on three parameters, q, V, and a, b u t for the purpose of determining %, we can tie t h e m into two dimensionless coefficients, A = q~Z/2K2, B = V/a~/2, (17) and calculate z~ t h r o u g h ~ from - . :c p ( q ( ) d ( ,
(18)
in which
pq(~) d~ = [1 + (A~2/2)] exp (-~2/4) sinh (B~) d~ 7r1/2 erf (B) exp (B 2) (1 + A + 2 A B : ) + 2 A B (19)
127
and the dimensionless variable ~ is related to z b y z = (A/q) ~2 = (a2/2K2) ~2.
Table I lists ~c for a n u m b e r of values of A and B. The limits selected for B cover all the combinations of V and a in Fig. l, while the range in A is chosen so t h a t , in addition, the perihelion distance can be varied at least between 1 and 3AU. The m e d i a n semimajor axis can now be established for an a r b i t r a r y choice of V and a o f the p o p u l a t e d area of Fig. 1 b y i n t e r p o l a t i n g ~c in Table I and applying (20). F o r q within 1.2AU and several s t a n d a r d combinations
TABLE
I
QUANTITY ~c, ]~EI.ATED TO THE MEDIAN SEMIMAJOR A x I s OF INTERSTEI, LAR COMETS. AS A FUNCTION OF COEFFICIENTS A AND B
A B l 0 -°'7 10 -o.4 10 - ° ' a 1 10 °'1 10 °.2 10 °'a 19 °'4 l0 ° s
10 - ~ ' °
10 -1"5
10 - l ' °
10 - ° ' s
1
10 ° ' s
101"°
101"5
1.70 1.77 1.92 2.31 2.70 3.27 4.07 5.11 6.43
1.74 1.81 1.97 2.38 2.78 3.38 4.20 5.25 6.57
1.84 1.92 2.10 2.56 2.99 3.61 4.43 5.45 6.73
2.06 2.16 2.37 2.86 3.30 3.89 4.66 5.62 6.85
2.34 2.45 2.66 3.13 3.54 4.08 4.78 5.70 6.89
2.52 2.62 2.82 3.27 3.65 4.15 4.83 5.73 6.91
2.59 2.69 2.89 3.32 3.68 4.18 4.85 5.74 6.92
2.61 2.71 2.92 3.34 3.70 4.19 4.86 5.75 6.92
TABLE
II
MEDIAN I~ECIrROCAL SEMIMAJOR AXIS (1/a)c FOR STANDARD COMBINATIONS OF SYSTEMATIC VELOCITY W AND DlSVERSION a (q ~< 1 . 2 A U ) V (kmsec -l)
a ( k m s e c -1)
10 10 15 15 15 20 20 20 20 30
l0 30 7 15 25 5 10 20 40 15
(l/a)c ( A U ) -1 --0.252 --2.55 --0.292 --0.650 --1.79 --0.475 --0.573 --1.31 --5.34 --1.46
(20)
V ( k m s e c -1) 30 30 40 50 50 70 70 100 175 300
a ( k m s e c -1) 30 50 40 30 70 40 60 75 100 125
(1/a)¢ ( A U ) -1 --3.51 --9.25 --6.86 --5.29 --20.2 --ll.1 --18.5 --32.6 --74.8 --168
] 2N
ZDENEK SEKANINA 7
7
----7
•
[
(10,10)
,/"
(20.5) / (!5,7) (15351
(20,10) 130, t5) "'
(20
oO1
'=
/
(50,30)
0q(l/o) (iU)
(40,40)
o~
i
OOl
I
1_ . 2
i
.
.
.
.
.
.
ORIGINAL RECIPROCAL SEMIMAJOR AXIS,
Fl(i. within ~ i i l i l" ]lo}ioll })q(lia)
1 ~
I/o
4
(AU) ,I
2. l ) i s t r i l ) u t i o n tJ,~(lla) (d" r(,cil)roclit s(,l/liiiillj(>l > llx(,s (~f Jnt(~rst('llar (~oin('ts w i t h p e r i l i e l i a l . g A U f]'oni i, ho ~illi. T h ( ' t w o i~]711l'(~s lit cacti Olli'V(~ I2i\:(' tllo s y s t o n m t i e v o l ( ) e i i y relat i\'e t() l lie a n d t i m v o h ) c i t y disi)(,rsi(tn cs (t)()tli hi l~:nls(,o - i ), r('sl)(~ctiv('l.v. T i l e fi'a(;{iOli o f t.onlet>~ w i t h p(,ri(lista, n(,es q % 1 . 2 A U t l l a t h a v e r~!cipro('fll s(~ininlaj()r a x e s I)(,tween ]la a n d l l a + d ( l / ' a ) is
d( l /a).
o{' 1" and ~, which are marke<| in Fig. I by open circles, the median (l/a)c is listed in T a b l e I I . The
(21)
( ; = A ÷ ~B 2.
If (i <: 1, the curve decreases from the beginning: if(;== ~,1 it a t t a i n s a l n a x i m u m a t I l a = (): and if'(/:> 2,its m a x i n i u m t a k e s place at (l/a),,, z~ < o, which relates to ~,,, according to (20), where ~,, satisfies the e(luation t a n h (B~,.) B~,,l l + (A~.//:)I
I I + (~,,,->/2)1 l1 + ( A ~ , f - / 2 ) [
. ~4~,.:
(22) E v e r y piece of informat.ion gained from s t u d y i n g the theoretical distribution of s e m i m a j o r axes of interstellar comets points to the s a m e conclusion: ()n the ass u m p t i o n of a similarity in the dynanficaI b e h a v i o r between the eloud of these comets a n d one of the k n o w n classes of galactie objeets, the <)rbits of the o v e r w h e h n i n g m a j o r i t y of interstellar comets, if we saw
t h e m , would be st,rongly hyperbolic, with I / a *~ = 0. I ( A U ) - ' ; i.e., t h e y would d e v i a t e ti'om a p a r a b o l a b v at ]cast 3 orders of magnitude more a p p r e c i a b l y t h a n does the most hyperbolic c o m e t a r y orbit aetually observed. The results obtained for q # 1.2 AU show only a s l i g h t - t o - m o d e r a t e depen
The a p p a r e n t absence of interstellar comets a m o n g tile o b s e r v e d comets can be used, following Eq. (13), to establish an upper limit on the space c o n c e n t r a t i o n of interstellar comets in tile solar neighborhood. In order to have figures readily coral)arable with those available on other classes of objects in the G a l a x y , it is useful to convert the space concentration N 0 into the space density P0 of interstellar comets. E x p r e s s i n g the latter in solar masses per cubic parsee, we have, fi'om (13), Po = g l ( q ) / c q
F(q),
(23)
where il~(q) is the actual mass influx, in g r a m s per century, of interstellar comets with periheli<)n less t h a n q, c :- 3 ,~, ] ¢p9, q
ENCOU:NTERS
WITH
INTERSTELLAR
is in AU, a n d F(q) is in A U k m s e c - l . To express P0 in g c m -3, its value in MGpc -3 m u s t be multiplied b y a f a c t o r of 6.77 x 10-23. To establish an u p p e r limit on P0, we det e r m i n e first the c h a r a c t e r of the dependence of expression (23) on t h e s y s t e m a t i c v e l o c i t y V relative to the Sun a n d on t h e v e l o c i t y dispersion a. I t can easily be shown t h a t p e a k space densities of interstellar comets for an a r b i t r a r y choice of M(q) a n d q are located along a c u r v i n g " r i d g e , " a t t a i n i n g the m a x i m u m v a l u e of Pmax =
~f(q)
(~/2c/¢q3/2)
-1
(24)
a t the p o i n t V = Vo = K(2/q) 1/2, a ~ O, a n d slowly sloping d o w n to P'0 = 0r/2) 1/2 3;/(q) (2cKq3/2) -1 = (rrl/2/2)Pmax
(25)
at the p o i n t a = a o = Kq - 1 / 2 : VolVO-2, V - + 0. The d e n s i t y d r o p s r a t h e r steeply on b o t h sides of the ridge, b u t g r a d u a l l y levels off a t V >> V0 a n d ¢ >> %. The actual m a s s influx M can be determ i n e d as the m a s s influx of discovered comet.s p divided b y their m e a n discovery p r o b a b i l i t y G. B o t h q u a n t i t i e s d e p e n d not only on perihelion distance, b u t also on the intrinsic brightness I 0. F u r t h e r m o r e , the influx t~ is a p r o d u c t of the discovery r a t e 8 (comets per century) a n d the m e a n m a s s of the discovered comets_M. Thus, in total, ;if(q) =: 3(q, AIo) M(q, AIo)[G(q, A / o ) ] - ' , (26) where q indicates the m a x i m u m perihelion distance considered, while A I o s t a n d s for the considered r a n g e of the l u m i n o s i t y distribution. F i r s t of all, as a p p a r e n t l y no interstellar c o m e t was discovered during the p a s t 100 y e a r s or so (since which time, fairly precise orbits h a v e been available for p r a c t i c a l l y e v e r y o b s e r v e d comet), we adopt, as a v e r y c o n s e r v a t i v e u p p e r limit, 8 < 1 interstellar c o m e t per c e n t u r y regardless of q a n d A I o. N e x t , to assess M, we a s s u m e t h a t the d i s t r i b u t i o n of c o m e t masses is g o v e r n e d b y a p o w e r law M -~ d M , where Mml n <
] 29
COMETS
M < Mma x a n d s is a constant. F o r Mma x >> Mmin, the m e a n c o m e t m a s s implied is :
[(s
1)/(2
__
s)]
2 -as x M msi - ~n 2~I m
(s # l, 2).
(27)
Several a t t e m p t s were m a d e r e c e n t l y to d e t e r m i n e the e x p o n e n t s or a related p a r a m e t e r . E v e r h a r t ' s (1967) results indicate t h a t the intrinsic d i s t r i b u t i o n of longperiod comets varies as l o 5/2 d I o for comets b r i g h t e r t h a n absolute m a g n i t u d e 5.3 and as 1o s/3 d1 o for f a i n t e r ones. W i t h the massl u m i n o s i t y relation p r o p o s e d b y W h i p p l e (1975), these laws give s = 2 a n d s = 13/9, respectively. Using " n u c l e a r " m a g n i t u d e s of comets, R o e m e r (1968) f o u n d that. the c u m u l a t i v e n u m b e r of comets varies in inverse p r o p o r t i o n to the 1.5 p o w e r of the nuclear size, which corresponds to s = ~. F r o m V s e k h s v y a t s k y ' s catalog of t o t a l absolute m a g n i t u d e s , W h i p p l e (!.975) obt a i n e d a law l o 2"1 dlo, s = 1.73, for n e a r l y parabolic comets w i t h q < 1.2 A U o b s e r v e d during the period 1850-1949, e s t i m a t i n g t h a t the m a x i m u m m a s s for a c o m e t a r y b o d y is a b o u t Mma x ~ 1022g. The m i n i m u m m a s s Mini n is d e t e r m i n e d b y a limit imposed on the discovery p r o b a b i l i t y t;. E v e r h a r t (1967, T a b l e I I ) lists G as a function of perihelion distance a n d absolute m a g n i t u d e . F o r q < 1.2AU, G does not d r o p below a b o u t 0.2 for comets b r i g h t e r t h a n absolute m a g n i t u d e about. 7-7.5, for which W h i p p l e ' s m a s s - l u m i n o s i t y relation gives Mmi n ~ 1015g. Thus, we find f r o m (27) t h a t the m e a n m a s s M in our case is 10~S'Sg w i t h R o e m e r ' s d i s t r i b u t i o n law; l 017.3 g w i t h W h i p p l e ' s ; and, from a modified form of (27), 1016"6g with E v e r h a r t ' s . A t the s a m e t i m e we find from L v e r h a r t s T a b l e I I t h a t ~7 > 0 . 3 for these comets. I n s e r t i n g into (26) the respective limits for $_and G, plus the g e o m e t r i c m e a n of the t h r e e M ' s , we get a reasonable u p p e r limit on the actual m a s s influx of interstellar comets : 3)/(1.2) < 1 0 1 S g p e r c e n t u r y .
(28)
W e note t h a t the limits o f q a n d I 0 h a v e been t a k e n r a t h e r arbitrarily. W e a d h e r e d to the limit on q picked u p b y Whipple, which allowed us to use some of his results w i t h o u t introducing a n y corrections. W e
I
3(i)
ZDENEK S E K A N ] N A
p o i n t o u t , h o w e v e r , that, w e w o u l d n o t g a i n a n y t h i n g f r o m s t r e t c h i n g q to m u c h larger values, b e c a u s e o f t h e r a p i d l y (leereasing d i s c o v e r y p r o b a b i l i t y at. large q ( E v e r h a r t , 1967). Similarly, e x t e n d i n g the lum inosity d i s t r i b u t i o n t o w a r d f a i n t e r c o m e t s woul(t n o t result in a b e t t e r v a l u e o f 3.)r, p a r t l y again, b e c a u s e o f a low d i s c o v e r y proba.bilitv o f faint comets, ~md p a r t l y also because o f t h e s u b s t a n t i a l increase in the unc e r t a i n t y e l M due to t h e error in t h e exp o n e n t s. A d o p t i n g 3:~ f r o m (2S) an(t
fbr l" ~ 0, ~ ~ 0, this limit r e m a i n s on t h e o r d e r o f m a g n i t u d e o f a few t i m e s 10 -4 M pc -~ for ~dl w d u e s of" I" a n d (~ t h a t could possibly be considered. These figures confirm %Thipple's (1975) r e c e n t result d e r i v e d f r o m a simplified t h e o r y . H o w e v e r , since we h a v e been v e r y c o n s e r v a t i v e in c h o o s i n g the n u m e r i c a l v a l u e s for t h e q u a n t i t i e s s e t t i n g p . . . . a m o r e realistic u p p e r limit m i g h t well be an o r d e r o f m a g n i t u d e l o w e r t h a n (29), an(t, o f course, t h e a,c t u a l spa(:(, (lensity o f interstellar c o m e t s may. still [)e m a n v orders o f m a g n i t u d e h)wer t h a n t h a t . In t e r m s o f t h e space c o n c e n t r a t i o n , t h e m a x i m u n l d e n s i t y w o u l d i n d i e a t e the ln'esenee of" o11(, eolnet p e r si)her(, 1 2 A [ r in ra,di
V. A CLOUD OF [NTERSTEI,LAR (IOMETS T R A V E L I N G WITH THE S[:N
Pmax <: (J X 10 -4 M ,l)e -3 : : 4 ::< I 0 -'¢'Kcm --3.
(2:)) or a b o u t ( ) . 5 % o f t h e t o t a l d e n s i t y o f t h e m a t t e r in the solar n e i g h b o r h o o d . E x c e p t
I
j
us.
T h e case o f zero s y s t e m a t i c v e l o c i t y o f a cloud relative to the S u n h a s n o t been covere(1 explicitly in t h e p r e v i o u s sections. T h e general t h e o r y f o r m u l a t e d in Sections
UPPER LIMIT ON i (10-4M° PC-5)
____
/ ::.\
\
/ i
20
/a we
i
'0
20
40
60
80
I00
G (kin sec-I)
FH'. 3. Upper limit on the spa,ee density ofinter.~tellm'conu'ts with pcriheliawithin 1.2AU a~ a flmction of th(' velocity dispersion cr and th(, system~ti(- \eloeity 1" relative to the Sun. Not.(, tht. m~ximum al cr=(L 1"=38.4kms~'(: ~. and th(" "ri(bze" M~)ping (l()wn t() or= 27.2kms('( '-I. I ' = 0 . whm',, tim limit on the sl)a(.~' d(,~J~ily ('¢lual 0.8.q ,)f'lh(, ]D~IXilIIIIIIIx.'IINHL
ENCOUNTERS
WITH INTERSTELLAR
I I and I I I gives the following relations for V-~0. /¢2 (
p~(z)dz=-~o2
and, for q .> K2/O"z or A/>- 1,
z,, = 2(az/• z - l/q)
O.2) --1
l+q~
× exp ( - h
(34) Our calculations for interstellar comets traveling with the Sun are s u m m a r i z e d in Table I I I . Unlike in Section I I I , we now would e x p e c t most interstellar comets to have nearly parabolic orbits, as long as the velocity dispersion in the comet cloud is k e p t low, well u n d e r l k m s e c -1. The space density implied is t h e n of order 10-SM+ pc -3 or less ( < 1 0 - 2 7 g c m - 3 ) . The present formulation of the problem (neglecting solar a t t r a c t i o n on the comets) does not provide for a capture-prone or a S u n - b o u n d comet cloud even in the case V = 0, a -~ 0. H o w e v e r , because of the low S u n - c o m e t velocities involved, our idealized approach is no longer adequate, and this case becomes a problem of an 0 o r t - t y p e cloud, where solar and stellar p e r t u r b a t i o n s m u s t be t a k e n into account. Unless ~ - + 0, however, Table I I I indicates t h a t strongly hyperbolic orbits result overwhelmingly, even in the case of zero systematic velocity V, and t h a t the u p p e r limit on the space density of interstellar comets is t h e n practically identical with the estimates derived from the general t h e o r y (Section IV).
lqz)
z) dz,
(30)
(+-)
× exp - ~ z
= 1 - Ill ÷ 1 (1 + 2A) -~qz] × exp ( - h z )
PO = ~
,
(31)
/¢2 ÷ qo.'-'''~
_ ~r(q) (~A) '/~ (1 + 2 A ) - ' , cscq 3/2 {2
-2Alog¢{211+a(1 7
= (2/q) (2A - 1).
(l + ½qz)
q (1 + 2A) -~ (1 + 4A
2cr2
] 31
COMETS
(1
(32)
O'2~--1
+ 2 A ) -~qzc] },
(33)
TABLE
11I
( I / a ) DISTRIBUTION PARA=~IETERS AND SPACE--DENSITY ESTIMATES lrOR INTERSTELLAR COMETS (q ~ 1 . 2 A U ) TRAVELrX(' ~VITH THE SUN
(W : 0 ; SOLAR ATTRACTION NEGI,ECTED)
( 1/a)~
( 1/a ),.
( k i n s e e -1 )
(AU) -l
(AU) -l
0.1 0.2 ().5 1 2 5 ]0
--0.0000156 --0.0000625 --0.000391 --0.00157 --0.00629 --0.0404 --0.176
--------
~10 ~10 <10 <10 <10
--
<10 -3.7
<10 -26.3 < 1 0 -25,9
--
< 10 -3"4
< 1 tl - 2 5 . 6
21)
--0.895
--
< 1 0 -3.3
<10 -25.5
--0.363 --3.97 --9.38
<10 -3.3 < 10 - 3 . 3 <10 -3.4
<10 -25.5 < 10 -2s-5 <10 -25.6
30 50 70
--2.43 --8.12 --17.1
po (M+pc -z ) -4.s -4.s -4.7 -4.4 -4"1
(gem -3 ) ~10 -27.°
~ 1 0 -26"7 < 1 0 -26.9 <10 -26.6
132
ZDENEK SEKANINA V[. (:(>N(JLUSIONS
()bserw~tional evidence virtually rules out the existence, in the solar neighhorhood, of a cloud of interstellar comets with a s])ace density of 6 × l 0 -4 solar masses l)er cuhic parsec ( 4 x lo-><~gcm 3) or higher and makes the existence of a conlet cloud l(i times less dense v e r y unlikely. However, the probability of e n c o u n t e r with the Sun of a comet from a cloud whose space density does not exceed l0 ~ solar masses per cubic parsec (hess than 10 e7gcm 3) is too low to exchlde a priori the, existence of interstellar comets~dtogether. For comparison, we note t h a t O o r t ' s (1950) estimates of the size a n d p o p u l a t i o n of his model of a solar-bound comet cloud give a nlean sl)ace (lensity of only a b o u t l0 -6 solar masses per cubic parsec (~10-_>8gcm 3), although Whipple (1975) has recently shown t h a t the possible unobserved mass of comets in the solar system could be a not-insignificant fra(..tion of the solar mass. The hypothetical interstellar comets might have originated in. s i l u o r might have be(m, at birth, s t a r - b o u n d comets t h a t were later ejected from circumstellar areas in a m a n n e r similar, perhaps, to the expulsion of a fl'action of nearly parabolic comets fronl the solar system b v p l a n e t a r y lierturhations. I n e i t h e r case, the m o t i o n s o f such interstellar colnets should sh(m an a|)liF('ciable s y s t e m a t i c velocity relative to the Sllli (II1OI'(. ~ t h a n l()kmsee
i) an(t a (lisper-
sion in in(lividual veh)cities of at least 5knlse( :--~, judging fl'oln analogy with other groups of galactic objects. While a (listinct systenlatic motion of the cloud relative to the Sun implies the Iiresence of a iweferential direction in the distribution of eonletary aphelia, which was detected I)y some (e.g., Tyror, 1957 ; H urnik, 195(.)) in the motions of nearly l)ar~boli(: comets, it also re(tuires t h a t an overal)un(tance o f s t r o n g l y h y p e r b o l i c o r b i t s be observed, w h i c h , o f course, has never materialize(I. ()n the o t h e r hart(l, the a p p a r e n t al)sen(;e
()fstrongly hyl)erbolic orl)its, together wilh the scarce occtlrrence o f s l i g h t l y hvI)er
holh: orbits alnollg the ohserve(l ('Olllets~ c()uhl only ilnl)ly a clou(l of comets travelilig w i t h the Sun (no s y s t e n l a t i e velocity
and a very low dispersion), but the aplielion l)oints would then have to be distribute(t essentially isotropically. Since an alternative and v e r y plausible explanation ( n o n g r a v i t a t i o n a l effects) is at hand for the few slightly hyperbolic original orbits t h a i are observed, we do not find it, a t t r a c t i v e to link a n y of the orbital properties of the actual comets with the hypothesis of interstellar comets. Tim model considered here coul(I, of course, lie generalized even further by considering an ellipsoidal distribution of coinet velocities (11, i.e., by w i t h d r a w i n g restriction (2). In the light of our present results, howeve, r, such a. move suhstant i ally co in plicati ng the m at he in ati cal treatment of the p r o b l e m - - d o e s not a p p e a r to be worth the effort, not at least in the fi)reseeable future. A('KNOWI,EDGMENT T h i s r(,search ~ a s mipport(,(| I)y ~_,rant N8(~ 7t)b12 fr()lii t h e Nat:toned A e r o n a u t i c s ~liid Sioaot~ Adlniliis(,rlt I i()ll. |{EFERENCES AI,I,I+]N, (I. li~'. (19731.
A,vtrophy,s'ical Qna~.titie,~'.
3rd od. Alhhmo Press, London.
Db:LttAVF, J. (19651. Solar mot,ion and velocit.v distributior~ of (.mm.m stars. In Ualactie Ntr,ct,,re (A. I~hlaux~ and M. Selimidt, Fds.), pp. t;l ~44. University of (',hiealzo lh'0ss, Chiea~'o. EVEltHAI~T, E. (19(;7). Intrinsic distrilmlions ~d' eolnctary poriholia and nla~nitmh's. , t Mro/#.,]. 72. 1002 1011. HVI:BSFI~, 1,'V. F., SN,, I)ER, 1,. E., AND I'll:Ill,, D. (1974). HCN r0x|io (,nlission frolll (~oiiilq I{oltoutol-: (1973f). learu.s '~3, 580 584. iluI~NIK, H. (19591. The distmibution of the directions of porihcdia, an(I of the orbital p, dos ( )f' n o i i .| )l q.il )d i(~ f~o n v.,l s, .q cla.4,~;t~"olt,. 9, 2t ) 7 ")_21.
?tlARSI)I.ZN. B. (I-., ANI) ~EKANINA. Z. f l 9 7 3 ) . ()it th(' d i s t r i b u t i o n o f "ori~ziiml" ()l'])ItS
Ills
1124.
~'IAI'~SliEN. I/. (L. ~I.]KANINA, Z., AND YEOMANS.
1), K. (19731. (loinets mid ll.nffravitatiollal f'or('cs V. Astro~l.J.78, 211 225. ()owr..I. 11. (19501. The st,ruct,uro of the clolld of (,I)lil('ls s l l r r o t i n d i l i ~ t h e solar s y s l o i n , a n d it [i3'l)l)lh(~sis (!OilOtWllili 7" it, s o r i g i n . 11.11. A.s'tr
ENCOUNTERS WITH INTERSTELLAR COMETS RICHTER, N. B. (1963). T h e N a t u r e of Comets. Methuen, London. ROEI~fER, E. (1968). Dimensions of the nuclei of periodic and near-parabolic cornets (abstract). Astron. J . 73, $33. TYROR, J. C~. (1957). The distribution of the directions of perihelia of long-period comets. M o n . Not. Roy. Astron. Soc. 117, 370-379.
] 33
ULICH, B. L., AND CONKLIN, E. K. (1974). Detection of methyl cyanide in Comet Kohoutek. N a t u r e 248, 121-122.
WHIPPLE, F. L. (1975). Do comets play a role in galactic chemistry and y-ray bursts? A stron. J . 80, 525 531.